cs3102: Theory of Computation Class 11: Moore, Mealy, and Markov Models Spring 2010 University of Virginia David Evans Menu Exam Review Variations on DFAs: Moore Machine: states produce output Mealy Machine: edges produce output Markov Model: transitions have probabilities
Moore Machine Edward Moore, Gedanken-experiments on Sequential Machines, 1956. http://people.mokk.bme.hu/~kornai/MatNyelv/moore_1956.pdf Moore Machine Example 0 q0; 1 1 0
1 q1; 0 Power of a Machine Power of a DFA, NFA, DPDA, NPDA/CFG: Set of languages it can recognize/produce. Power of a Moore Machine: Set of functions it can perform. Language Set of strings Function Set of
(input/output) pairs Formal Definition Computing Model DFA Moore Computing Model DFA Moore
Moores Experiments Okay...guess the machine! You LOSE! 0 q0; 1 1 0
0 0 q1; 0 q2; 0 1 1 1
0 q6; 1 q3; 0 1 0 q5; 0
1 q4; 0 1 0 You always lose. Sometimes you win... Lorenz Cipher Machine
used by Nazi high command: links between conquered capitals Machine determined by Bill Tutte (1941) from intercepted messages Colossus Arguably, the first electronic, digital, programmable computer. Bletchley Park, 1943
Decoded 63 million letters in Nazi command messages Learned German troop locations to plan D-Day (knew the deception was working) Bletchley Park, 2004 (rebuilt) A More Fair Game Reveal: n, maximum number of states in the machine (and , input alphabet) Equality Rule: two machines are the same if they compute the same function
= {0, 1} n = 3 0 0 q1; 0 q2; 0 1 q3; 1
1 0 How many experiments is enough? Alternate Game Given: state machine Experiment: input -> output Win: guess what state the machine started in Moore proved for some machines where all states are distinguishable, it is impossible to know the starting state from one experiment.
Mealy Machine George Mealy, A Method for Synthesizing Sequential Circuits, 1955 0; 1 q0 1; 0 0; 1 1; 0 q1 Mealy
Machine Moore Machine Computing Model Mealy Machine Moore Machine
Computing Model Which is more powerful? Moor e Mealy For any Moore Machine M, we can construct a Mealy Machine M that performs the same function: qb; y qa; z
qi; x For any Moore Machine M, we can construct a Mealy Machine M that performs the same function: qb qb; y qa qa; z
x x qi; x qi For any Mealy Machine M, we can construct a Moore Machine M that performs the same function: qb qa y x
qi For any Mealy Machine M, we can construct a Moore Machine M that performs the same function: qb qb qa qa y x
qi qi1; x qi2; y Both have all the same outgoing transitions as qi Equally Powerful Moor e
Mealy (Moore may need more needs more states) Are they good models? 0; 1 q0 0; 1 1; 0 1; 0
q1 Markov Model 1.0 0.3 Happy Grumpy 0.2
0.4 0.7 0.3 Sleepy 1 0. Sneezy 0.1 0.9
Andrey Markov, 1856-1922 Markov Model with1.0Outputs 0.3 ARRGH Grumpy 0.3 0.7 0.7 Happy
0.3 Sleepy 0.5 1.0 0.5 ho ho ho! Sneezy 0.1
wahoowa! 0.9 1.0 achoo! #%#$& Zzzzzzzz Markov Model Examples
Nodes: URLs Links: hyperlinks Probabilities: 1/n number of nonself outgoing links a.com b.com 3 1/ 1/3
1/2 c.org 3 1/ 1/2 Pr(u) = probability of reaching u starting from random seed states d.com
1/2 1/2 Lawrence Page, Sergey Brin, Rajeev Motwani and Terry Winograd Garkov http://www.joshmillard.com/garkov/ Hidden Markov Model 1.0 0.3
0.4 Happy 0.3 0.7 0.7 0.3 Sleepy
0.5 1.0 1 0. 0.5 ho ho ho! Sneezy 0.1
wahoowa! ARRGH Grumpy 0.2 0.9 1.0 From just the outputs guess the states (and machine)
achoo! #%#$& Zzzzzzzz Hidden Markov Model Example No Review Question No Review Question No Review
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Lazy 0.9 1.0 No Review Question 0.1 Sent Review
Topics Active Student 1.0 Hidden Markov Model AA Flop: 222 Opponent Raises 0.4
0.6 0.8 AK 0.9 72 0.02 0.08 Raise
Call Fold Return PS3 front of room A-D E-K L-R
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